What is the highest integer power of 6 that can divide 73! – 72! ?

Question

What is the highest integer power of 6 that can divide  73! – 72!  ?

A) 14 
B) 16 
C) 34 
D) 36 
E) 68

Devesh26 · Answer

Basically he is asking how many 6 are there in the equation. On simplifying it becomes 72*72!. There are 12 6 multiples in 72!. In 36 multiples contain 2 sixes so 36 and 72 contain 4 sixes and out of these 4, 2 are already counted in 6 multiples so we should consider only 2 sixes. And finally the 72 in 72*72! Contains 2 sixes. So totally 12+2+2=16

So I think the answer is 16

Posted from my mobile device

So I think the answer is 16

Posted from my mobile device

vipulgoel · Answer

we got to check the highest power of 3 in equation 73!-72 bcz combination of 2*3 gives us 6 and we can get highest power of 2 as many as 3 so we will check for highest power of 3 in the stem

72!72

gives 24+8+2 and two 3 for 72 = 36 threes

Ertucinar · Answer

73! - 72! = 73 	imes 72! - 1 	imes 72! = (73 - 1) 	imes 72! = 72 	imes 72! 
 72! 	imes 72 = 2^3 	imes 3^2 	imes 72!

Highest Power of 6
Since  6 = 2 	imes 3 , we need the  minimum power of 2 and 3  in  72! 	imes 72 .

- Highest power of 2 in  72! :

Using Legendre’s formula:

\sum_{k=1}^{\infty} \left\lfloor \frac{72}{2^k} ightfloor = \left\lfloor \frac{72}{2} ightfloor + \left\lfloor \frac{72}{4} ightfloor + \left\lfloor \frac{72}{8} ightfloor + \left\lfloor \frac{72}{16} ightfloor + \left\lfloor \frac{72}{32} ightfloor + \left\lfloor \frac{72}{64} ightfloor

= 36 + 18 + 9 + 4 + 2 + 1 = 70

Including the extra  2^3  from 72:

70 + 3 = 73

- Highest power of 3 in  72! :

\sum_{k=1}^{\infty} \left\lfloor \frac{72}{3^k} ightfloor = \left\lfloor \frac{72}{3} ightfloor + \left\lfloor \frac{72}{9} ightfloor + \left\lfloor \frac{72}{27} ightfloor

= 24 + 8 + 2 = 34

Including the extra  3^2  from 72:

34 + 2 = 36

The highest power of 6 is determined by:

min(73, 36) = 36

Option D - 36

ravi1522 · Answer

Hello ,
73!−72! =72!(73-1)=72!*72
so 72! will have maximum power of 34 of 6
72 will also have max power of 2 as 72 can written as 6*6*2

so total maximum power will be 36
Hence D is correct answer
I hope it Helps

Thank you !!

What is the highest integer power of 6 that can divide 73! – 72! ?

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What is the highest integer power of 6 that can divide 73! – 72! ?

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